LECTURE NOTES ON QUANTUM CHAOS
COURSE 18.158, MIT, APRIL 2016, VERSION 2
SEMYON DYATLOV
Abstract. We give an overview of the quantum ergodicity result.
1. Quantum ergodicity in the physical space
1.1. Concentration of eigenfunctions. First, let us consider the case when M ⊂ R2
is a bounded domain with piecewise C ∞ boundary and we take the operator
∆ = −∂x21 − ∂x22 .
We study the Dirichlet eigenvalues λ0 < λ1 ≤ λ2 ≤ . . . (with multiplicities taken into
account) and the corresponding L2 normalized eigenfunctions uj ∈ H01 (M ), so
∆uj = λ2j uj ,
uj |∂M = 0,
kuj kL2 (M ) = 1.
(1.1)
We are interested in the following question regarding the high energy limit:
Question 1.1. How do uj concentrate as j → ∞?
In general, uj become rapidly oscillating at high energies, so we have to study their
concentration in some rough sense. A natural way to do that is to take the weak limits
of the measures |uj (x)|2 dx along subsequences:
Definition 1.2. Let ujk be a subsequence of (uj ) and µ a probability measure on M .
We say that ujk → µ weakly, if
Z
Z
2
a(x)|ujk (x)| dx →
a(x) dµ(x) for all a ∈ C0∞ (M )
(1.2)
M
M
We say that ujk equidistributes in M , if it converges weakly to the volume measure:
ujk →
dx
.
Vol(M )
(1.3)
Remarks. 1. By a standard density argument, once (1.2) holds for all a ∈ C0∞ (M ),
it holds for all a ∈ C(M ).
2. By a diagonal argument (see [Zw, Theorem 5.2]), there always exists a subsequence
of uj converging to some measure.
1
2
SEMYON DYATLOV
As a basic example, consider the square
M = [0, 1]2 .
The Dirichlet eigenfunctions have the form
uj` (x1 , x2 ) = 2 sin(jπx1 ) sin(`πx2 ),
j, ` ∈ N;
λj` = π
p
j 2 + `2 .
Exercise 1.3. Show that as j → ∞,
ujj → dx1 dx2 ;
u1j → 2 sin2 (πx1 ) dx1 dx2 ,
that is the sequence ujj equidistributes in M but the sequence u1j does not.
More generally, we will consider the case when (M, g) is a compact Riemannian
manifold with piecewise smooth boundary and replace ∆ by the Laplace–Beltrami
operator ∆g which can be defined using the identity
Z
Z
(∆g u)v d Volg , u, v ∈ C0∞ (M ).
hdu, dvig d Volg =
M
M
The generalization of the measure (1.3) to this case is given by the Riemannian volume
measure
d Volg
.
Volg (M )
Exercise 1.4. Let M = S2 be the two-dimensional sphere embedded into R3 .
(a) Using the expression for Laplacian on R3 in spherical coordinates, show that each
homogeneous harmonic polynomial v on R3 of degree m, the restriction u := v|S2 is an
eigenfunction of ∆S2 with eigenvalue m(m + 1). (In fact, with a bit more work one can
see that all eigenfunctions of ∆S2 are obtained in this way.)
(b) Using the coordinates (x1 , x2 , x3 ) in Rn , define for each m ∈ N0 ,
±
vm
= (x1 ± ix2 )m ,
±
u±
m := cm vm |S2 .
where the constant cm is chosen so that ku±
m kL2 (S2 ) = 1. Show that as m → ∞,
±
um converge weakly to a probability measure on S2 which is supported on the equator
{x21 + x22 = 1, x3 = 0}.
We see that the limit of ujk may depend on the choice of the sequence. It turns out
that the limits in fact also depend in an essential way on the dynamics of a natural
flow on (M, g), and quantum chaos studies in particular how the dynamical properties
of (M, g) influence the behavior of eigenstates.
LECTURE NOTES ON QUANTUM CHAOS
3
1.2. Chaotic dynamics and ergodicity. For (M, g) a Riemannian manifold, define
the unit cotangent bundle
S ∗ M = {(x, ξ) ∈ T ∗ M : |ξ|g = 1}.
When M is a domain in R2 , we can write
S ∗ M = {(x, ξ) ∈ M × R2 : |ξ| = 1}
and parametrize this space by (x1 , x2 , θ) where ξ = (cos θ, sin θ). (The unit tangent
and cotangent bundles can be identified with each other using the metric g, and it
will become apparent later why it is much more convenient for us to use the cotangent
bundle here.)
We consider the geodesic billiard ball flow on M ,
ϕt : S ∗ M → S ∗ M,
t ∈ R.
For M a domain in R2 , every trajectory of ϕt follows a straight line with velocity vector
ξ until it hits the boundary, when it bounces off by the law of reflection. For (M, g) a
Riemannian manifold, straight lines are replaced by geodesics induced by the metric g.
If M has a boundary, then the resulting map is not continuous and it is defined
everywhere except a measure zero set in R × S ∗ M , corresponding to trajectories that
either hit non-smooth parts of the boundary or become tangent to the boundary.
We will ignore these issues in our note and send the reader to [ZeZw] for a detailed
explanation of how they can be handled.
A natural probability measure on S ∗ M is the Liouville measure, defined for a general
Riemannian manifold by
dµL =
d Volg (x)dµSn−1 (ξ)
,
Volg (M ) · Vol(Sn−1 )
n = dim M,
where µSn−1 is the standard surface measure on the sphere, transported to a measure
on each fiber of S ∗ M . For M a domain in R2 , in coordinates (x1 , x2 , θ) we have
dµL =
dx1 dx2 dθ
.
2π Vol(M )
The measure µL is invariant under the flow:
µL (ϕt (U )) = µL (U ),
U ⊂ S ∗ M,
t ∈ R.
We now introduce the notion of ergodicity for the flow ϕt , which is a rather weak way
of saying that ϕt is a chaotic flow:
Definition 1.5. We say that ϕt is ergodic with respect to µL , if for each flow invariant
set
U ⊂ S ∗ M ; ϕt (U ) = U, t ∈ R,
we have either µL (U ) = 0 or µL (U ) = 1.
4
SEMYON DYATLOV
One important consequence of ergodicity is the following statement about ergodic
averages
Z
1 T
a ◦ ϕt dt, T > 0, a ∈ L1 (S ∗ M ; µL ).
(1.4)
haiT :=
T 0
Theorem 1 (L2 ergodic theorem). Assume that ϕt is ergodic with respect to µL . Then
for each a ∈ L2 (S ∗ M ; µL ),
Z
haiT →
a dµL in L2 (S ∗ M ; µL ).
S∗M
Proof. We will only sketch the proof, sending the reader to [Zw, Theorem 15.1] for
an alternative proof, and we restrict ourselves to the case when M has no boundary.
Consider the vector field X on S ∗ M generating the flow, so that
ϕt = exp(tX).
This vector field gives rise to a first order differential operator, still denotes X. Since
µL is a ϕt -invariant measure, we have LX µL = 0 and thus −iX is an unbounded
self-adjoint operator on L2 (S ∗ M ).
Let dEX be the spectral measure of −iX, which is an operator-valued measure on R
which is constructed via the spectral theorem for unbounded self-adjoint operators.
Then for a ∈ L2 (S ∗ M ; µL ),
Z
eitλ dEX (λ)a.
a ◦ ϕt = exp(tX)a =
R
Therefore, for T > 0
Z haiT =
R
Now the function
limit
eiT λ −1
iT λ
1
T
Z
T
e
itλ
0
Z iT λ
e −1
dt dEX (λ)a =
dEX (λ)a.
iT λ
R
is bounded uniformly in T, λ, and it has the pointwise in λ
eiT λ − 1
→ 1l{0} (λ) =
iT λ
(
1, λ = 0;
0, λ 6= 0,
as T → ∞.
Since integral over the spectral measure is a strongly continuous function of the interval,
one can see from here that
Z
haiT →
dEX (λ)a in L2 (S ∗ M, µL ).
(1.5)
{0}
The right-hand side is the orthogonal projection of a onto the space V0 ⊂ L2 (S ∗ M, µL )
of functions satisfying the equation Xf = 0. However, for each such f we have
f ◦ ϕt = ϕt and thus the sublevel sets {f ≤ c} are invariant under the flow (modulo
a measure zero set which can be removed). By ergodicity, V0 must then consist of
LECTURE NOTES ON QUANTUM CHAOS
5
constant functions. Then the right-hand side of (1.5) is the integral of a with respect
to µL , finishing the proof.
Exercise 1.6. Show that neither [0, 1]2 nor S2 have ergodic ϕt . (Hint: on S2 , the
angular momentum with respect to any axis gives a conserved quantity. Any sublevel
set of this function will be invariant under the flow.)
There are many important examples of ergodic systems, including
• Sinai billiards;
• Bunimovich stadiums;
• Riemannian manifolds (M, g) without boundary which have negative sectional
curvature, in particular closed negatively curved surfaces.
1.3. Statement of quantum ergodicity. The following theorem (together with its
generalizations is Theorems 4, 8 below) is the main result to be proved in this course:
Theorem 2 (Quantum ergodicity in the physical space). Assume that ϕt is ergodic
with respect to µL . Then there exists a density 1 subsequence λjk , that is
#{k | λjk ≤ R}
→1
#{j | λj ≤ R}
as R → ∞,
such that ujk equidistributes in M :
ujk →
d Volg
.
Volg (M )
This theorem was stated by Shnirelman [Sh] and proved by Zelditch [Ze] and Colin
de Verdère [CdV]. The case of the domains with boundary was established by Zelditch–
Zworski [ZeZw]. See [Zw, Theorem 15.5] for a detailed proof in the boundaryless case.
(All of the results mentioned above prove the more general Theorems 4,8.)
We see that Theorem 2 uses information about the cotangent bundle on M to derive
a statement on the manifold M itself. It turns out that to prove it, we should generalize
the statement of equidistribution to T ∗ M , which we call the phase space.
2. Phase space concentration and proof of quantum ergodicity
We will henceforth assume that M has no boundary, referring the reader to [ZeZw]
for the boundary case.
2.1. Semiclassical quantization. Assume that a ∈ C0∞ (T ∗ M ). Semiclassical quantization associates to a, which is called symbol or classical observable, an operator
Oph (a) : L2 (M ) → L2 (M )
6
SEMYON DYATLOV
which is called a semiclassical pseudodifferential operator or quantum observable. This
procedure depends on a parameter h > 0, called the semiclassical parameter, and we
will be interested in the limit h → 0. Originally h referred to (a dimensionless version
of) Planck constant; in general it is the wavelength at which we want to study our
eigenfunctions.
We will not give a definition of Oph (a) here but will instead send the reader to [Zw,
Chapters 4 and 14], and will give some explanations regarding semiclassical quantization later in the course. We remark that the procedure is independent of the choice
of coordinates on M only modulo an O(h) remainder in the symbol, but the defined
class of operators is geometrically invariant.
In fact we may define Oph (a) for a in a more general class S m (T ∗ M ), m ∈ R, given
by the conditions
a ∈ S m (T ∗ M ) ⇐⇒ |∂xα ∂ξβ a(x, ξ)| ≤ Cαβ (1 + |ξ|)m−|β| .
The resulting operator acts on Sobolev spaces
Oph (a) : H s (M ) → H s−m (M ),
We note that if a(x, ξ) is a polynomial in ξ,
X
a(x, ξ) =
aγ (x)ξ γ ,
s ∈ R.
aγ (x) ∈ C ∞ (M ),
|γ|≤m
then Oph is a differential operator; on Rn , the standard quantization procedure gives
X
1
Oph (a) =
aγ (x)(hDx )γ , Dx = ∂x .
(2.1)
i
|γ|≤m
In particular, if a(x, ξ) = a(x), then we get a multiplication operator
Oph (a)u(x) = a(x)u(x),
and on Rn ,
Oph (ξj ) = hDxj =
h
∂x .
i j
Also, if X is a vector field on M , then we have
h
X = Oph (pX ) + O(h), pX (x, ξ) = hξ, X(x)i.
i
This explains why our symbols are functions on the cotangent bundle rather than the
tangent bundle – a vector field naturally gives a linear function on the fibers of the
cotangent bundle.
We list below some fundamental properties of the quantization operation. We
leave the remainders ambiguous, but they will have appropriate mapping properties in
Sobolev spaces.
LECTURE NOTES ON QUANTUM CHAOS
7
Theorem 3. For a ∈ S m (T ∗ M ), b ∈ S k (T ∗ M ), we have
Oph (a)∗ = Oph (ā) + O(h),
Oph (a) Oph (b) = Oph (ab) + O(h),
h
Oph ({a, b}) + O(h2 ),
i
where {a, b} is the Poisson bracket, given in coordinates by
X
{a, b} =
(∂ξj a∂xj b − ∂xj a∂ξj b).
[Oph (a), Oph (b)] =
(2.2)
(2.3)
(2.4)
j
Moreover, for a ∈ S 0 (T ∗ M ) the operator norm of Oph (a) on L2 can be estimated as
follows [Zw, Theorem 5.1]: for some constant C independent of a, h,
lim sup k Oph (a)kL2 →L2 ≤ CkakL∞ (T ∗ M ) .
(2.5)
h→0
2.2. Quantum ergodicity in phase space. We now generalize Theorem 2 to a
statement about quantum observables
hOph (a)uj , uj iL2 ,
a ∈ S 0 (T ∗ M ).
For that we need to pick the value of h and it will be convenient to put
hj =
1
.
λj
We then define
Vj (a) := hOphj (a)uj , uj iL2 (M ) ,
a ∈ S 0 (T ∗ M ).
Definition 2.1 (Weak limits in phase space). Let ujk be a subsequence of uj and µ be
a measure on T ∗ M . We say that ujk → µ in the sense of semiclassical measures, if
Z
Vjk (a) →
a dµ for all a ∈ S 0 (T ∗ M ).
T ∗M
We say that ujk equidistribute in phase space if they converge to the Liouville
measure:
ujk → µL .
Remark. There is always a subsequence converging to some measure, and all resulting
measures are supported on the unit cosphere bundle S ∗ M and invariant under the flow
ϕt – see [Zw, Chapter 5] and (2.8), (2.10) below.
Theorem 4 (Quantum ergodicity in phase space). Assume ϕt is ergodic with respect
to µL . Then there exists a density 1 subsequence ujk such that ujk equidistribute in the
phase space.
8
SEMYON DYATLOV
Theorem 2 follows from here by taking a to be a function of x, so that
Z
a(x)|uj (x)|2 dx,
Vj (a(x)) =
M
and using the fact that the pushforward of µL to M is the volume measure:
Z
Z
1
a(x) dµL =
a(x) d Volg .
Volg (M ) M
S∗M
In the rest of this section, we prove Theorem 4, following several steps.
2.3. Step 1: using the eigenfunction equation. We first rewrite the eigenfunction
equation
∆g uj = λ2j uj
in the form
Ophj (p)uj = 0
(2.6)
where the symbol
p(x, ξ) = p0 (x, ξ) + O(h),
p0 (x, ξ) =
|ξ|2g − 1
2
(2.7)
is chosen so that
P = Oph (p) =
h2 ∆g − 1
.
2
Define the Hamiltonian vector field
X
Hp0 =
∂ξj p · ∂xj − ∂xj p · ∂ξj ,
j
and note that
Hp0 a = {p0 , a},
a ∈ C ∞ (T ∗ M ).
Define the flow
ϕt = exp(tHp0 ),
then (explaining the choice of
the geodesic flow.
1
2
in the definition of p0 ) the restriction of ϕt to S ∗ M is
A key tool in the proof is the Schrödinger propagator
itP U (t) = U (t; h) = exp −
: L2 (M ) → L2 (M ).
h
It quantizes the flow ϕt as made precise by the following
Theorem 5 (Egorov’s Theorem). For a ∈ C0∞ (T ∗ M ), we have
U (−t) Oph (a)U (t) = Oph (a ◦ ϕt ) + O(h)L2 (M )→L2 (M ) .
LECTURE NOTES ON QUANTUM CHAOS
9
Proof. We only sketch the proof, see [Zw, Theorem 15.2] for details. It is enough to
prove that, denoting at := a ◦ ϕt ,
∂t U (t) Oph (at )U (−t)) = O(h)L2 (M )→L2 (M )
The left-hand side is
i
ih
U (t) Oph (∂t at ) − P, Oph (at ) U (−t)
h
By (2.4), this becomes
U (t) Oph (∂t at − {p0 , at })U (−t) + O(h)
and it remains to use that ∂t at = {p0 , at }.
We then have the following
Lemma 2.2. Assume that a ∈ C0∞ (T ∗ M ). Then for any T > 0,
Vj (a) = Vj haiT + OT (hj )
where haiT is defined in (1.4) and the constant in the remainder depends on T .
Proof. We have for each t, U (t; hj )uj = uj and thus
Vj (a) = hOphj (a)uj , uj iL2 = hU (−t; hj ) Ophj (a)U (t; hj )uj , uj iL2
= hOphj (a ◦ ϕt )uj , uj iL2 + Ot (hj ) = Vj (a ◦ ϕt ) + Ot (hj )
and it remains to average both sides over t ∈ [0, T ].
(2.8)
This statement uses the fact that uj are eigenfunctions and features ergodic averages
along the flow ϕt .
2.4. Step 2: basic bounds. We record here a few standard bounds on Vj (a). First
of all, by (2.5) we have for some global constant C and each a ∈ S 0 (T ∗ M ),
lim sup |Vj (a)| ≤ CkakL∞ (T ∗ M ) .
(2.9)
j→∞
Moreover, if a vanishes on S ∗ M , then
lim Vj (a) = 0
j→∞
(2.10)
as follows immediately from
Lemma 2.3 (Elliptic bound). Assume a ∈ S 0 (T ∗ M ) and a|S ∗ M = 0. Then as j → ∞,
k Ophj (a)uj kL2 = O(hj ).
10
SEMYON DYATLOV
Proof. Since a vanishes on S ∗ M , we may write a = bp0 = bp + O(h), where p, p0 are
defined in (2.7) and b ∈ S −2 (T ∗ M ). Then by (2.3),
Ophj (a) = Ophj (b) Ophj (p) + O(hj )L2 →L2 .
Since Ophj (p)uj = 0 by (2.6), the proof is finished.
2.5. Step 3: bounding averages over eigenfunctions. We know by Theorem 1
that for large T , the average haiT is close to the integral of a, but only in L2 (T ∗ M ).
If we had an L∞ estimate instead, then we could use (2.9) to control Vj (a) for all j in
the limit j → ∞. However, ergodic averages typically do not converge in L∞ (this can
be seen for instance by considering a closed geodesic).
Therefore we will have to make the best out of the L2 bound on a. It turns out that it
produces a bound on Vj (a) on average in j – see Lemma 2.4 below. The key statement
is the following theorem, which we will try to prove later in the course (see §3.3):
Theorem 6 (Local Weyl Law). Assume that χ ∈ C0∞ (0, ∞) and a ∈ S 0 (T ∗ M ).
Then as R → ∞,
R n Z
X λj ξ −1
Vj (a) =
χ |ξ|g a x,
dxdξ + O(R ) .
χ
R
2π
|ξ|g
T ∗M
j
Taking a = 1 in Theorem 6, we in particular get
X λj R n Z
χ
χ |ξ|g dxdξ + O(R−1 ) .
=
R
2π
T ∗M
j
Approximating χ = 1l[0,1] by functions in C0∞ (0, ∞) , this gives
Theorem 7 (Weyl Law). We have as R → ∞,
ωn
Volg (M )Rn + o(Rn )
#{j | λj ≤ R} =
n
(2π)
where ωn > 0 is the volume of the unit ball in Rn .
Note also that the integral on the right-hand side in Theorem 6 is zero if a vanishes
on S ∗ M ; this is in line with Lemma 2.3.
For the proof of quantum ergodicity, we use the following corollary of Theorem 6:
Lemma 2.4 (Variance bound). We have for each a ∈ S 0 (T ∗ M ), as R → ∞
Z
X
−n
2
R
|Vj (a)| ≤ C
|a|2 dµL + O(R−1 )
λj ∈[R,2R]
S∗M
where the constant C depends on M , but not on a or R.
LECTURE NOTES ON QUANTUM CHAOS
11
Proof. Take nonnegative χ ∈ C0∞
estimate
X λj X λj χ
χ
k Ophj (a)uj k2L2 = R−n
hOphj (a)∗ Ophj (a)uj , uj iL2
R−n
R
R
j
j
(0, ∞) with χ = 1 on [1, 2], then it is enough to
and the right-hand side is bounded by Theorem 6 using that by (2.2) and (2.3)
Ophj (a)∗ Ophj (a) = Ophj |a|2 + O(hj ) = Ophj |a|2 + O(R−1 ). 2.6. Step 4: integrated quantum ergodicity. We can now prove the following
integrated (or, strictly speaking, summed) form of Theorem 4:
Theorem 8 (Integrated quantum ergodicity). Assume that a ∈ S 0 (T ∗ M ) and
Z
a dµL .
(2.11)
La =
S∗M
Then as R → ∞,
R−n
X
|Vj (a) − La |2 → 0.
λj ∈[R,2R]
Proof. By subtracting La from a and using that Oph (1) is the identity operator, we
reduce to the case La = 0:
Z
a dµL = 0.
S∗M
Moreover, by (2.10) we may assume that a ∈ C0∞ (T ∗ M ).
Take some T > 0. By Lemma 2.2 and then Lemma 2.4, we have
X
X Vj haiT 2 + OT (R−1 )
R−n
|Vj (a)|2 ≤ R−n
λj ∈[R,2R]
λj ∈[R,2R]
≤
CkhaiT k2L2 (S ∗ M,µL )
+ OT (R−1 )
where the constant C is independent of T and R. Taking the limit as R → ∞, we have
X
lim sup R−n
|Vj (a)|2 ≤ CkhaiT k2L2 (S ∗ M,µL ) .
R→∞
λj ∈[R,2R]
The left-hand side does not depend on T , and the right-hand side converges to 0 by
Theorem 1.
2.7. Step 5: end of the proof. It remains to derive Theorem 4 from Theorem 8,
that is to extract a density 1 sequence of eigenfunctions which equidistributes in phase
space. For that we use Chebyshev inequality and a diagonal argument on dyadic pieces
of the spectrum.
More precisely, for r ∈ N let
Nr := #{j | λj ∈ [2r , 2r+1 )},
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SEMYON DYATLOV
then Nr ∼ 2nr as r → ∞ by the Weyl law (Theorem 7). Take a sequence
as ∈ C0∞ (T ∗ M ),
s = 1, 2, . . .
which is dense in C0∞ (T ∗ M ) with respect to the uniform norm. Put Ls =
and
X
1
2
ε`,r := max
|Vj (as ) − Ls | .
s≤`
Nr
r r+1
λj ∈[2 ,2
R
S∗M
as dµL
)
Then ε`,r → 0 as r → ∞ for each ` by Theorem 8. We pick r(`) such that r(`+1) > r(`)
and
ε`,r < 100−`
for r ≥ r(`).
Define the disjoint collection of sets J` ⊂ N as follows:
j ∈ J` ⇐⇒ λj ∈ [2r(`) , 2r(`+1) ) and max |Vj (as ) − Ls | < 2−` .
s≤`
By Chebyshev inequality, for r(`) ≤ r < r(` + 1),
# {j | λj ∈ [2r , 2r+1 )} \ J`
≤
`ε`,r
Nr ≤ 2−` Nr ,
2−2`
therefore
1−
#(J` )
≤ 2−` .
# {j | λj ∈ [2r(`) , 2r(`+1) )}
It follows from here and the Weyl law that the sequence
[
J`
jk , {jk } =
`
is a density one subsequence in N.
On the other hand, we have for each s,
Z
Vjk (as ) →
as dµL
as k → ∞.
S∗M
Using the bound (2.9) and the fact that {as } is dense in C0∞ with the uniform norm,
we see that
Z
Vjk (a) →
a dµL as k → ∞
S∗M
for all a ∈ C0∞ (T ∗ M ). By (2.10) same is true for all a ∈ S 0 (T ∗ M ), finishing the proof.
3. Overview of semiclassical quantization
We now briefly discuss how to define the quantization procedure Oph , sending the
reader to [Zw] for details.
LECTURE NOTES ON QUANTUM CHAOS
13
3.1. Quantization on Rn . We consider the following symbol classes on T ∗ Rn = R2n ,
Shm (T ∗ Rn ) ⊂ C ∞ (Rn ),
m ∈ R,
defined as follows: a(x, ξ; h) ∈ S m (T ∗ Rn ) if for each multiiindices α, β there exists a
constant Cαβ such that for all x, ξ and small h,
|∂xα ∂ξβ a(x, ξ; h)| ≤ Cαβ (1 + |ξ|)m−|β| .
Note that for m ∈ N0 this class includes polynomials of order m in ξ with coefficients
bounded with all derivatives in x.
For a ∈ Shm (T ∗ Rn ), we define the operator Oph (a) on functions on Rn as follows:
Z
i
−n
Oph (a)f (x) = (2πh)
e h hx−y,ξi a(x, ξ)f (y) dydξ.
(3.1)
R2n
The integral (3.1) does not always converge in the usual sense, so some explanations
are in order. Assume first a is smooth and compactly supported, or more generally a
lies in the Schwartz class S (T ∗ Rn ). If f ∈ S (Rn ), then integral in (3.1) converges
absolutely and gives a Schwartz function.
When a ∈ S m (T ∗ Rn ), we see using the semiclassical Fourier transform
Z
i
−n/2
Fh f (ξ) = (2πh)
e− h hy,ξi f (y) dy
Rn
that
−n/2
Z
Oph (a)f (x) = (2πh)
i
e h hx,ξi a(x, ξ)Fh f (ξ) dξ
(3.2)
Rn
and since Fh f (ξ) is Schwartz, the integral still converges; integrating by parts in ξ, we
see that it still gives a Schwartz function.
In fact, for any a ∈ S m (T ∗ Rn ), one can define Oph (a)f for f ∈ S 0 (Rn ), where
S 0 (Rn ), the dual to S (Rn ), is the space of tempered distributions. This can be seen
either by duality or by treating (3.1) as an oscillatory integral, or by first considering
the case of a ∈ S (T ∗ Rn ) and extending to general a by density. In either case, we
obtain the quantization procedure on Rn ,
a ∈ Shm (T ∗ Rn ) 7→ Oph (a) : S (Rn ) → S (Rn ), S 0 (Rn ) → S 0 (Rn ).
Moreover, rapidly decaying symbols produce smoothing operators:
a ∈ S (T ∗ Rn ) =⇒ Oph (a) : S 0 (Rn ) → S (Rn ).
Note that the mapping properties above are for any fixed h; we make no statement
about the uniformity of norms as h → 0 at this point.
Exercise 3.1. Using (3.2), show that when a ∈ S m (T ∗ Rn ) is polynomial in ξ, the
operator Oph (a) is the differential operator defined in (2.1).
14
SEMYON DYATLOV
In what follows, we will often ignore what happens as x, ξ → ∞, so our proofs
would immediately work for Schwartz symbols a ∈ S (T ∗ Rn ) and with more work can
be extended to general symbols.
3.2. Basic properties of quantization and stationary phase. We now want to
establish some properties of the quantization procedure Oph . We start with the product
formula (2.3). Assume that a, b ∈ S (T ∗ Rn ) uniformly in h. We would like to write
Oph (a) Oph (b) = Oph (c),
c(x, ξ; h) ∈ S (T ∗ Rn ),
(3.3)
and understand the asymptotics of c as h → 0.
We first find a formula for c using the following statement, known as oscillatory
testing; see [Zw, Theorem 4.19]:
Lemma 3.2. Assume that a ∈ S (T ∗ Rn ). Then for each fixed h > 0,
1. We can recover the symbol a from the operator A = Oph (a) as follows:
i
i
a(x, ξ) = e− h hx,ξi A(e h h•,ξi ).
(3.4)
2. If A : S 0 (Rn ) → S (Rn ) and the function a ∈ S (T ∗ Rn ) satisfies (3.4), then
A = Oph (a).
We now write out the symbol c from (3.3) as follows:
i
i
c(x, ξ) = e− h hx,ξi Oph (a) Oph (b)(e h h•,ξi )
i
i
= e− h hx,ξi Oph (a)(b(•, ξ; h)e h h•,ξi )
Z
i
−n
= (2πh)
e h hx−y,η−ξi a(x, η; h)b(y, ξ; h) dydη.
(3.5)
Rn
To understand the behavior of c as h → 0, we use the following (see [Zw, Theorem 3.16])
Theorem 9 (Method of stationary phase). Assume U ⊂ Rn is an open set and ϕ ∈
C ∞ (U ; R) has only one critical point x0 ∈ U , that is ∇ϕ 6= 0 on U \ {x0 }.
Assume also that x0 is a nondegenerate critical point, that is the Hessian ∇2 ϕ(x0 )
gives a nondegenerate quadratic form. Denote by sgn(∇2 ϕ(x0 )) the signature of this
form (the number of positive eigenvalues minus the number of negative eigenvalues).
Then for each a ∈ C0∞ (U ; C), we have as h → 0
Z
∞
iϕ(x0 ) X
iϕ(x)
n/2
h
h
e
a(x) dx ∼ (2πh) e
hj Lj (a)|x=x0
U
(3.6)
j=0
where each Lj is a ϕ-dependent linear differential operator of order 2j. In particular
−1/2
iπ
2
L0 (a)|x=x0 = e 4 sgn ∇ ϕ(x0 ) det ∇2 ϕ(x0 )
a(x0 ).
LECTURE NOTES ON QUANTUM CHAOS
15
Proof. We only sketch a proof in a special case known as quadratic stationary phase:
n = 1,
ϕ(x) =
x2
.
2
By Fubini’s Theorem and a linear change of variables, one can pass from here to the
case when ϕ is a nondegenerate quadratic form in higher dimensions. The general
case can then be handled by the Morse Lemma, which gives a change of variables
conjugating a general phase ϕ locally to a quadratic form.
We compute in terms of the standard (nonsemiclassical) Fourier transform â(ξ),
r Z
Z
ihξ2
ix2
iπ
h
e 2h a(x) dx = e 4
e− 2 â(ξ) dξ.
(3.7)
2π R
R
This follows from the more general statement true for any z ∈ C, Re z ≥ 0, z 6= 0:
Z
Z
2
ξ2
1
− zx2
a(x) dx = √
e− 2z â(ξ) dξ.
(3.8)
e
2πz R
R
The statement (3.8) follows for z > 0 by direct calculation using the Fourier transform
of the Gaussian and for all z by analytic continuation.
Now, taking the Taylor expansion of e−ihξ /2 as h → 0 and using that â ∈ S , we
get
r
Z
∞ Z
ix2
iπ
h X
1 ihξ 2 j
e 2h a(x) dx ∼ e 4
−
â(ξ) dξ
2π j=0 R j!
2
R
2
iπ
∼ e4
√
2πh
∞
X
1 ih j 2j
∂x a(0)
j!
2
j=0
finishing the proof.
In the case (3.5) we integrate over y, η, thus the dimension is 2n. The phase is given
by
(y, η) 7→ hx − y, η − ξi,
and the only critical point is y = x, η = ξ. The value of the phase at the critical
point is equal to 0. The expansion (3.6) can be computed explicitly from quadratic
stationary phase and yields
n
hX
c(x, ξ) ∼ a(x, ξ)b(x, ξ) +
∂ξ a(x, ξ)∂xj b(x, ξ) + O(h2 ),
i j=1 j
explaining (2.3), (2.4).
16
SEMYON DYATLOV
3.3. More on semiclassical quantization. On a manifold M , we define the quantization Oph (a) by covering M with a locally finite system of coordinate charts, splitting a into pieces using a partition of unity, quantizing it separately on each chart
using (3.1), and adding the pieces back together. However, if we take different charts
or the partition of unity, the resulting operator will change by an operator with symbol
in hShm−1 (T ∗ M ).
Therefore, it is more convenient to consider the class of semiclassical pseudodifferential operators
Ψm (M ) = {Oph (a) | a ∈ Shm (T ∗ M )}
which is independent of the choice of quantization, and the principal symbol map
σh : Ψm (M ) → Shm (T ∗ M )/hShm−1 (T ∗ M ),
σh (Oph (a)) = a + hShm−1 (T ∗ M )
which is also independent of the quantization. We have the short exact sequence
σ
h
0 → hΨm−1 (M ) → Ψm (M ) −→
Shm (T ∗ M )/hShm−1 (T ∗ M ) → 0.
The symbolic calculus makes it possible to construct more pseudodifferential operators by calculating their symbol term by term. For instance, we can find approximate
inverses of operators with nonvanishing symbols:
Proposition 3.3. Assume a ∈ Sh0 (T ∗ M ), p ∈ Shm (T ∗ M ), and p 6= 0 on supp a. Then
there exists b ∈ Sh−m (T ∗ M ) such that
Oph (a) = Oph (b) Oph (p) + O(h∞ ).
Remark. This generalizes Lemma 2.3 in the following sense: if (h2 ∆g − 1)u = 0, then
k Oph (a)ukL2 = O(h∞ )kukL2
for all a ∈ Sh0 (T ∗ M ), supp a ∩ {|ξ|g = 1} = ∅.
Sketch of proof. We first take
b0 =
a
∈ Sh−m (T ∗ M ).
p
Then by (2.3) we have for some r1 ∈ Sh−1 (T ∗ M ),
Oph (a) = Oph (b0 ) Oph (p) + h Oph (r1 ) + O(h∞ ).
Moreover, one can arrange so that supp r1 ⊂ {a 6= 0}. Then we repeat the procedure,
putting
r1
b1 =
∈ hSh−m−1 (T ∗ M ).
p
Arguing this way we construct some symbols bj ∈ hj Sh−m−j (T ∗ M ) and it remains to
take b such that
X
b∼
bj . j
LECTURE NOTES ON QUANTUM CHAOS
17
We can also take functions of pseudodifferential operators, which we present in a
special case. Namely we have
Theorem 10. Assume (M, g) is a compact Riemannian manifold without boundary,
and put P = h2 ∆g = Oph (p) + O(h)Ψ1h where p(x, ξ) = |ξ|2g . Then for each χ ∈ C0∞ (R)
and all N , we have
χ(P ) ∈ Ψ−N
σh (χ(P )) = χ(p).
h (M );
Sketch of proof. We write using the Fourier transform χ̂,
Z
1
χ(P ) =
χ̂(t)eitP dt.
2π R
(3.9)
For bounded t, we have
eitP = Oph (pt ),
pt = eitp + O(h),
(3.10)
as can be done solving the equation
∂t Oph (pt ) = iP Oph (pt )
in symbolic calculus. When χ̂ is compactly supported, we get the desired formula.
Otherwise we can write (3.10) up to t ∼ hε for some small ε, where the symbol pt will
have derivatives mildly growing in h, and use the integral (3.9) with the fact that χ̂ is
Schwartz.
From Theorem 10 we can derive the following version of local Weyl law of Theorem 6
(the original version, with h depending on uj , can be proved using a rescaling and
Lemma 2.3):
Theorem 11. For χ ∈ C0∞ (R), a ∈ S 0 (T ∗ M ), and λj , uj defined in (1.1), we have as
h→0
Z
X
2
−n
χ(h λj )hOph (a)uj , uj i = (2πh)
χ(p(x, ξ))a(x, ξ) dxdξ + O(h1−n ).
j
T ∗M
Proof. Putting P := −h2 ∆g , the left-hand side is the trace
tr χ(P ) Oph (a) .
However, we know that χ(P ) Oph (a) ∈ Ψ−N
h (M ) for all N , so we write χ(P ) Oph (a) =
Oph (b) + O(h∞ ) where b = χ(p)a + O(h) is rapidly decreasing in ξ.
It remains to use the following trace formula for pseudodifferential operators:
Z
−n
b(x, ξ) dxdξ + O(h1−n )
tr Oph (b) = (2πh)
T ∗M
which reduces to the case of quantization on Rn and there the trace can be computed
by integrating the Schwartz kernel.
18
SEMYON DYATLOV
3.4. Another application of stationary phase: concentration of Lagrangian
states. Assume that U ⊂ Rn is open and we are given a phase function ϕ ∈ C ∞ (U ; R)
and an amplitude b ∈ C0∞ (U ; C). We define the family of functions uh ∈ C0∞ (U ; C),
h > 0, by
uh (x) = eiϕ(x)/h b(x).
We would like to understand the limits as h → 0 of observables
hOph (a)uh , uh iL2 ,
a ∈ C0∞ (T ∗ Rn ),
specifically to write for some measure µ,
Z
hOph (a)uh , uh iL2 →
a dµ.
(3.11)
T ∗ Rn
This can be done by applying the method of stationary phase to
Z
i
−n
Oph (a)uh (x) = (2πh)
e h (hx−y,ξi+ϕ(y)) a(x, ξ)b(y) dydξ.
R2n
The phase function is
(y, ξ) 7→ hx − y, ξi + ϕ(y),
the stationary point is given by
x = y,
ξ = ∇ϕ(x),
and the value of the phase at the stationary point is equal to ϕ(x). Applying (3.6), we
obtain
Oph (a)uh (x) = eiϕ(x)/h a(x, ∇ϕ(x))b(x) + O(h).
Therefore we have the limit (3.11) with µ given by
Z
Z
a dµ =
a(x, ∇ϕ(x))|b(x)|2 dx.
T ∗M
U
In particular, µ lives on
Λϕ = {(x, ∇ϕ(x)) | x ∈ U }
which is a Lagrangian submanifold of T ∗ Rn .
Exercise 3.4. Find the semiclassical limits (in the sense of Definition 2.1) of the
functions u1j and ujj from Exercise 1.3. (Ignore the boundary issues by testing these
functions against operators supported strictly inside the square.)
LECTURE NOTES ON QUANTUM CHAOS
19
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